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It’s the Friday Puzzle…..

There are 2 flagpoles that are each 100 foot high. A rope that is 150 feet long is strung between the tops of the flagpoles. At its lowest point the rope sags 25 feet about the ground (see schematic diagram below). How far apart are the flagpoles?

As ever, please do NOT post your answers, but do say if you think you have solved the puzzle and how long it took. Solution on Monday.

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

Wait…
So I spend about five minutes looking at quadratic equations, getting nowhere, then look at the comments for a hint and see that people are getting it very quickly.
Then I read the question again.
That’s… rather silly, actually. One day I’ll learn that the more mathematical the Friday puzzle looks, the less it probably is.

I heard the puzzle on Car Talk, an American radio program, several months ago. It took me a couple of minutes while I was listening to the rest of the show and walking to work to get it.

So… either a couple of minutes while busy with other things, negative several months, or as long as it took me to read enough of the question to verify I’d heard it before and all the particulars matched, say 10 seconds. Take your pick.

This was pretty easy, and lack of mathematical skill was actually a benefit to me, I think. Instead of starting with some preconceived notion of how to solve the geometry, I just started assembling everything I knew and could easily discern from the information available, and the answer immediately became obvious. Then I spent some time trying to figure out how I must be wrong.

I started out thinking about how to do an integral in calculus and then started thinking about how I could use a Newtonian approximation before thinking about how I could just do a linear approximation and then the solution became very very obvious. I might have overthought this one.

Like several others I started out trying to work out how to work out the solution. At first I tried using an approximation to get some rough numbers. That’s when I realized my rough answer was the real answer. Total time about 5 minutes.

Read the question, thought about picking up my old textbook in maths, thought about how hard (impossible?) this would be for people who have not done maths at this level. Read the question again trying to find an answer wich did not require any skills at math. Got it in about 15s total, while beeing slightly intoxicated. (guess I fail at life for spending 5 min writing a message online on a friday night while intoxicated)

Not really, you know: while in a homogenous gravitational field all catenaries are similar to each other, I doubt that’s still the case in a radial field. Further, we need to know the azimuth of the line connecting the flagpoles, their latitudes, and the day length, all to account for the influence of centripetal forces. Shame that this of course makes symmetry fly out the window (in a strictly Newtonian manner I’m sure).

Well, you raise an interesting point: things are already complicated enough when considering just the Newtonian effects, but what about relativistic ones? We’d need to know not only the precise planetary gravity gradient, but also the distance from the primary and its characteristics, as well as those of any other large masses in the system. Then there are quantum effects, and don’t forget Brownian motion of the atmosphere!

I started reading the comments before I thought clearly about the problem, and was happy to read that I didn’t need to use catenaries coz I’ve never heard of them.
I think I’ve got the answer without facepalming too, though I did a bit of face slapping before I got it.

Thought it would involve parabola! Just did some maths, then thought parabola could also be ellipse. Solved for 5-10 minutes. Read comments, people reported 15 seconds solved! I admit to doing something complex. And there was the answer – 5 seconds.